Semifree Isovariant Poincaré Spaces and the Gap Condition

Abstract

We introduce the notion of a semifree isovariant $G$-Poincaré space, a homotopical notion interpolating between semifree closed smooth $G$-manifolds and the equivariant Poincaré spaces of [HKK24b]. It carries the additional structure of an equivariant Poincaré embedding of the fixed points of a semifree $G$-Poincaré space. Under suitable gap conditions on the codimension, we show that the space of isovariant structures on a semifree $G$-Poincaré space for a periodic finite group $G$ is highly connected, giving a useful construction tool for manifold structures on equivariant Poincaré spaces.

30 pages, comments welcome!